PHY862 Accelerator Systems Introduction to Accelerators Homework Peter N. Ostroumov Professor of Physics Michigan State University

Homework, p.1 What is the kinetic energy in units of both Joules and electron volts for an electron accelerated through a dc potential of 1 Megavolt? Find an expression for the fractional error when the nonrelativistic approximation for kinetic energy as a function of  is used. (a) At what values of  and  does the error in kinetic energy equal 1%? (b) To what kinetic energy does this correspond, for electrons and for protons? If the only nonzero components of the electromagnetic field in cylindrical coordinates are Er, Ez, and B, write the nonzero components of the Lorentz force for a particle of mass m and charge q moving along the z direction with velocity v. The rate of work done by a force F acting on a particle with velocity v is . Using the definition of the Lorentz force and the appropriate vector relationship, derive the expression for the rate of kinetic energy gain for a particle of charge q, and show that the magnetic force does not contribute. P.N. Ostroumov PHY862 "Aceclerator Systems" Fall 2018

Homework, p.2 Please answer the following questions for a proton traveling at a velocity of 0.9c. What is its momentum [GeV/c]? What is its kinetic energy [GeV]? What is its rigidity [T-m]? If this proton travels through a 1cm long magnet with a 1T field, by what angle will it be deflected [rad]? The Relativistic Heavy Ion Collider (RHIC) collides fully stripped gold ions (A=197, Z=79) at a total energy of E_coll=100 GeV/nucleon per beam. The circumference of each ring is 3834 m. Assume the rest mass of a gold ion is 197×0.93113 GeV. Calculate the revolution frequency of a particle at the injection energy of E_inj=10.5 GeV/nucleon, and at the storage energy of E_coll=100 GeV/nucleon. What is the change in revolution frequency for particles accelerated from E_inj to E_coll? If we assume that there are 192 identical dipoles per ring, each of length L= 10 m, what is the required dipole field in each at the collision energy of E coll? Read APS booklet “accelerators and beams” https://www.aps.org/units/dpb/news/edition4th.cfm P.N. Ostroumov PHY862 "Aceclerator Systems" Fall 2018